Colloquium

Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations. The SPT phases in free fermion systems, like topological insulators, can be classified by K-theory. However, it is not yet known what SPT phases exist in general interacting systems. In this talk, I will first present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases. Just as group theory allows us to construct 230 crystal structures in three dimensions, we find that group cohomology theory allows us to construct many interacting bosonic SPT phases. In my talk, I shall show how topological terms in the path integral description of the system can be constructed from nontrivial group cohomology classes, giving rise to exactly soluble Hamiltonians with explicit ground state wavefunctions. Next, I will discuss the generalization of the classifying scheme to interacting fermionic systems and a new mathematical framework – group supercohomology theory, which predicts a fermionic SPT phase that can neither be realized in free fermionic nor interacting bosonic systems.

Finally, I will briefly mention the deep relationship between SPT phases and chiral anomalies in high energy physics.

Everything around us, everything each of us has ever experienced, and virtually everything underpinning our technological society and economy is governed by quantum mechanics. Yet this most fundamental physical theory of nature often feels as if it is a set of somewhat eerie and counterintuitive ideas of no direct relevance to our lives. Why is this? One reason is that we cannot perceive the strangeness (and astonishing beauty) of the quantum mechanical phenomena all around us by using our own senses. I will describe the recent development of techniques that allow us to image electronic quantum matter directly at the atomic scale. As examples, we will visually explore the previously unseen and very beautiful forms of quantum matter making up electronic liquid crystals [1,2]; hybridized heavy-fermions [3,4]; topological-insulator surface states [5]; and high temperature superconductors [6,7]. We will discuss the implications for fundamental research, and also for advanced materials and new technologies, arising from the development and application of these novel techniques .

Renormalization is a principled coarse-graining of space-time. It shows us how the small-scale details of a system may become irrelevant when looking at larger scales and lower energies. Coarse-graining is also crucial, however, for biological and cultural systems that lack a natural spatial arrangement. I introduce the notion of coarse-graining and equivalence classes, and give a brief history of attempts to tame the problem of simplifying and "averaging" things as various as algorithms and languages. I then present state-space compression, a new framework for understanding the general problem. At the end, I present recent empirical results, in an animal social system, that show evidence for the coupling of scales: the reaction of coarse-grained facts about a system "downwards" to influence the microphysics. 

In (relatively) recent years some philosophers of physics have developed and advocated a new view about how to understand spatiotemporal structures posited in theories such as classical mechanics, relativistic theories and GR; it is called the 'dynamical approach' to spacetime (H. Brown 2005, Physical Relativity).  The dynamical approach (DA) holds that spacetime structure should not be thought of as conceptually prior to the laws of nature, or as constraining the forms that the laws may have. Instead, the DA approach says that the laws of nature are prior, and spacetime structures are no more than a reflection, or codification, of facts (especially symmetry facts) about the dynamical laws in our world. In my talk I will explore the motivations and arguments given in support of the dynamical approach, and raise doubts about whether they are coherent and compelling. Although no-one should come away from my talk with a perfect understanding of the nature of spacetime (or even just: spacetime as it appears in classical relativistic theories), I hope that all will come to appreciate the difficulty of deciding what even clear and mathematically well-understood physical theories really tell us about basic aspects of physical reality.

Quantum many-body systems ranging from a many-electron atom to a solid material are described by effective Hamiltonians which are obtained from more accurate Hamiltonians by neglecting or treating weak interactions perturbatively. Quantum complexity theory asks about the quantum computational power of such quantum many-body models for both practical as well as fundamental purposes. Three distinct computational classes have emerged within this framework: namely (1) classical Hamiltonians such as the Ising model, (2) sign-free or stoquastic Hamiltonians such as the transverse field Ising model, and (3) fully quantum Hamiltonians such as the Heisenberg model. Each class can be characterized by certain prototype universal Hamiltonians which can encode the physics of any other Hamiltonian in that class. We will show how this encoding is established through the use of perturbation theory via perturbative gadgets. We will discuss the technical expression of this classification in terms of the complexity classes NP, Stoquastic MA and QMA and the power of these Hamiltonians for performing quantum adiabatic computation.

Gravity in 1+1 dimension is classically trivial but, as shown by A. Polyakov in 1981, it is a non-trivial quantum theory, in fact a conformal field theory (the Liouville theory), and also a string theory. In the last decades many important results and connexions with various areas of mathematics and theoretical physics have been established, but some important issues remain to be understood. In this colloquium I shall focus on some recent developments and new questions on the relation between discrete and continuous 2 dimensional gravity, probabilities and stochastic processes, random fractal geometries and SLE curves.

The supermassive black hole in the centre of the Milky Way, Sgr A*, is an ideal target for testing the properties of black holes. A number of experiments are being prepared or conducted, such as the monitoring of stellar orbits, the search for radio pulsars or the recording of an image of the shadow of a event horizon. The talk puts these efforts in context with other tests of general relativity and its alternatives. I will also compare the approaches in the Galactic centre, while concentrating in particular on the prospects of using pulsars as probes, also in combination with event horizon imaging.

Quantum information theory has taught us that quantum theory is just one possible probabilistic theory among many others. In the talk, I will argue that this "bird's-eye" perspective does not only allow us to derive the quantum formalism from simple physical principles, but also reveals surprising connections between the structures of spacetime and probability which can be phrased as mathematical theorems about information-theoretic scenarios.